The Kaye effect: New experiments and a mechanistic explanation

Journal of Non-Newtonian Fluid Mechanics 273 (2019) 104165

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The Kaye effect: New experiments and a mechanistic explanation J.R.C. King∗, S.J. Lind Department of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Manchester M13 9PL, UK

a b s t r a c t The Kaye effect is a phenomenon whereby a jet of fluid poured onto a surface appears to leap on impact, rather than stagnate or coil as expected. Since it was first described in 1963, several authors have attempted to explain the mechanism by which the phenomenon occurs, although to date no complete explanation for the behaviour exists. Current evidence points towards the existence of an air layer between the jet and the heap which enables slip. We show that the Kaye effect does not occur in a vacuum, indicating that the air layer is crucial for the effect to occur. By use of control volume analysis we show that viscoelasticity plays a key role in the Kaye effect, and this role is two-fold. Firstly, viscoelasticity appears to increase air entrainment, and secondly, it reduces the pressure required to bend the jet, allowing a thicker air layer to be sustained. Shear thinning behaviour reduces this viscoelastic response. These findings provide new insight into a problem that has puzzled rheologists for over half a century.

1. Introduction When a jet of viscous liquid is poured onto a surface, it usually coils and forms a heap. The Kaye effect, first observed by Kaye [1] and still not fully understood, is the phenomenon wherein, under certain conditions, the jet rebounds and a secondary jet appears to leap out from the heap. Collyer and Fisher [2] showed that the leaping jet is a loop, formed as the jet slips on the heap. All the fluids in which the Kaye effect has been observed exhibit both shear thinning and viscoelastic behaviour, and Collyer and Fisher [2] postulated that the mechanism for slip was a shear thinning layer between the jet and the heap. Versluis et al. [3] supported this explanation, stating that viscoelasticity played no role. The Kaye effect was also considered by Majmudar et al. [4], who investigated the coiling of a viscous jet from a dynamical systems perspective, and saw the Kaye effect as a regime observed during the transition towards chaotic motion. Majmudar et al. [4] observed that elasticity does play a role in the Kaye effect, although they did not suggest a mechanism for the influence of elasticity. Recent experimental work by Binder and Landig [5] has inferred the existence of a thin air layer between jet and heap, on which the jet slides, contradicting the shear thinning explanation. High speed photographs and videos presented by Lee et al. [6] provide clear evidence of the existence of an air layer. Recent numerical simulations [7] also support this theory. Despite this, there is currently no clear consensus on the mechanism by which the jet slips, let alone the reason why the effect occurs in some fluids and not others. There are other instances of liquid jets leaping off other fluids or films, for example Versluis et al. [3] demonstrated a phenomenon whereby a jet of viscous liquid was made to rebound from a soap film in a stable manner. There are many fluids which can be bounced off a soap film, a phenomenon which, we argue, should not be defined as the Kaye

∗

effect. Similarly, Thrasher et al. [8] caused a viscous Newtonian jet to rebound from a reservoir of the same fluid by rotating the reservoir, and showed that the jet was separated from the reservoir by a thin layer of air. Thrasher et al. [8] agree that this behaviour does not constitute the Kaye effect, arguing that the Kaye effect only occurs in non-Newtonian fluids. As a consequence of the confusion over which instances of bouncing jets constitute the Kaye effect (and why), we propose the following definition as used herein: The Kaye effect is the bouncing of a jet of fluid poured onto a heap made of the same fluid, whilst neither jet nor heap are subject to any mechanical interventions. The entrainment of a thin air film by a liquid, and the subsequent slip of the liquid on the air film, are known to occur under certain conditions for jet and droplet impacts with solid substrates. The behaviour of a liquid jet impacting on a moving substrate has been studied by Moulson et al. [9,10]. Splash and deposition regimes were identified, and in [9] it was shown that the transition between these regimes is dependent on the ambient pressure, with splash suppressed at lower ambient pressures. An analogous result was found by Xu et al. [11], Driscoll and Nagel [12], who observed that the splashing of droplets impacting on a solid surface may be suppressed by reducing the ambient pressure. Through numerical and theoretical modelling Sprittles [13,14] provided a mechanism for splash suppression, showing that at lower ambient pressures, the dynamics of the air around the liquid-solid interface are altered. This paper has two aims. The first aim is to confirm that air is essential for the Kaye effect to occur. We show experimentally that the Kaye effect does not occur in a vacuum, supporting the theory and observations that the jet slips on a thin air layer. The second aim is to determine (through a simplified analytic model) the effect of fluid properties in creating and sustaining the Kaye effect. By consideration of the stresses, we provide a mechanistic explanation of the phenomenon, showing why the Kaye effect occurs in some fluids and not others. We show that viscoelas-

Corresponding author. E-mail addresses: [email protected] (J.R.C. King), [email protected] (S.J. Lind).

https://doi.org/10.1016/j.jnnfm.2019.104165 Received 1 March 2019; Received in revised form 13 September 2019; Accepted 17 September 2019 Available online 18 September 2019 0377-0257/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

J.R.C. King and S.J. Lind

Journal of Non-Newtonian Fluid Mechanics 273 (2019) 104165

Fig. 1. Suppression of the Kaye effect in a vacuum. pcrit ≈ 32kPa (absolute pressure).

ticity is key, and that shear thinning behaviour reduces the influence of the elastic stress. These results are relevant to the wider scientific community, as the Kaye effect is often used as an example of interesting and unexplained phenomena in popular science and outreach. More fundamentally however, we are now closer to a complete explanation of an everyday phenomenon that has puzzled and fascinated for over half a century. In Section 2 we present our experimental findings. In Section 3 we put forward a theoretical model for the Kaye effect, and in Section 4 we discuss our findings based on this model. Section 5 is a summary of our conclusions. Further details of the experiment are provided in Appendix A. A video showing the key result of our experiment is included in the supplemental material. 2. Experimental findings We conducted an experiment to observe the Kaye effect in a vacuum chamber. Details of the experiment are given in Appendix A. The chamber was initially at atmospheric pressure, and the Kaye effect was observed to occur approximately 70% of the time for a constant flow rate. As the air was evacuated from the chamber, the pressure dropped, and the proportion of time the Kaye effect was observed decreased, until a critical (absolute) pressure pcrit ≈ 32kPa, below which the Kaye effect was completely suppressed. When the pressure in the vacuum chamber was increased the Kaye effect started again, at approximately the same value of pcrit . This behaviour is shown clearly in the video included in the supplemental material. Fig. 1 shows the typical behaviour of the jet above and below pcrit . Above pcrit the Kaye effect occurs, and due to the angle of the surface onto which the jet falls, the effect is quite stable, sometimes lasting for several seconds. Below pcrit the Kaye effect is suppressed, and the jet simply coils, with the coiled heap gradually subsiding. We define tKaye /Δt as the proportion of time the Kaye effect is observed in a given sampling window Δt. A value of 1 corresponds to continuous Kaye effect over the sampling window, whilst 0 is complete suppression of the Kaye effect. In the following we use a sampling window of Δ𝑡 = 4𝑠. Fig. 2 shows the variation of tKaye /Δt with absolute pressure. The well defined critical pressure below which the Kaye effect does not occur is clear in Fig. 2. Qualitatively, this observation was highly repeatable: We observed this behaviour for a range of flow rates and jet heights, and for two types of shampoo. We did not find any fluids, jet heights or flow rates for which the Kaye effect occurred below pcrit ≈ 32kPa. The frequency of occurrence of the Kaye effect for pressures above pcrit showed some variation. The value of pcrit varied between samples, we believe due to sensitivities of the Kaye effect to the amount of air entrained in the shampoo. We were unable to find any relationship between pcrit and the experimental parameters. For all flow rates, fall heights and shampoo types tested (which exhibited the Kaye effect), we found pcrit ∈ [32kPa, 50kPa]. As the pressure is reduced at constant temperature (as in the present case, see Appendix A), the mean free path l in the air scales inversely with absolute pressure. If we take the Knudsen number as

Fig. 2. The effect of pressure on the occurrence of the Kaye effect. The vertical axis is the proportion of time the Kaye effect was observed, with 1 being continuous Kaye effect and 0 complete suppression of the Kaye effect. The data were obtained over three evacuation-repressurisation cycles.

𝐾𝑛 = 𝑙∕𝛿, with the air layer thickness 𝛿 ≈ 0.5𝜇m as in [6], we obtain Kn ≈ 0.1 at atmospheric pressure, and Kn ≈ 0.35 at 𝑝𝑐𝑟𝑖𝑡 = 32𝑘𝑃 𝑎. For 0.01 < Kn < 0.1, slip flow occurs, whilst for 0.1 < Kn < 1 the flow transitions towards free molecular flow [15]. We postulate that the value of pcrit is related to the pressure at which the mean free path in the air becomes comparable to the thickness of any entrained air layer. In short, the air layer has no further load bearing capacity, and cannot act as a lubricating layer to support the slipping jet. If the air layer were to be modelled, below pcrit the flow could not be described by the Navier-Stokes equation augmented with a slip boundary condition. We further investigate the air layer using a lubrication theory based model in Section 4.3. We have demonstrated that air is a crucial ingredient for the Kaye effect to occur. This supports the theory of [5,6], that the jet slips on a thin layer of air. Furthermore, it demonstrates that the shear-thinning slip layer theory of [2] is at best incomplete. Given the existance of the air layer, we believe the key question relating to the Kaye effect is now: Why does the Kaye effect occur for some fluids and not for others? In the remainder of this paper we attempt shed light on this question.

3. Mechanistic model In light of the debate in the literature over the role of elasticity [3,4], we develop a simplified control volume model of the quasi-steady Kaye effect, to separately isolate the effects of viscoelastic and shear thinning behaviour. A key feature of viscoelastic fluids is that they exhibit a memory of their strain history. A viscoelastic fluid subjected to a non-zero strain rate for a finite period of time will continue to carry a stress for some time (characterised by the relaxation time) after the strain rate returns to zero. In the following we consider a jet of incompressible viscoelastic liquid with a relaxation time 𝜆. A Newtonian fluid is represented by 𝜆 = 0.

J.R.C. King and S.J. Lind

Journal of Non-Newtonian Fluid Mechanics 273 (2019) 104165

Fig. 3. Schematic diagram of idealised steady state Kaye effect, and the coordinates used in our control volume analysis.

Consider a jet with constant flow rate undergoing the Kaye effect. A schematic diagram of this is given in Fig. 3. The primary jet emerges from a reservoir with orifice of radius R0 , and falls a height H before hitting the heap and bending. The centreline of the jet round the bend is assumed to have a constant radius of curvature R, through an angle 𝜙. The system is assumed to be in a quasi-steady state: the timescale for changes in the geometry of the flow is significantly greater than the timescale for fluid to pass from the reservoir to the secondary jet. The coordinate system x, y has origin at the centre of the reservoir orifice. The q, r, 𝜃 coordinate system follows the curve of the jet: q is a coordinate along the jet length, and r and 𝜃 are coordinates within jet, as shown in Fig. 3. The origin of q, r, 𝜃 is the centre of the jet at the start of the bend (at (𝑥, 𝑦) = (0, −𝐻 )). The unit vectors are ex , ey , eq , er and e𝜽 . We denote the start and the end of the bend A and B respectively. We assume that the jet has circular cross section for the entirety of its length. We also assume that the air layer provides perfect slip, such that the jet does not experience any friction as it slides on the heap. From this it follows that between A and B the radius of the jet is constant with value r0 , and the velocity does not vary with q. We define a control volume V with surface S as the section of the jet between A and B (i.e., (q, r, 𝜃) ∈ ([0, R𝜙], [0, r0 ], [0, 2𝜋])). The jet is incompressible, with density 𝜌, viscosity 𝜂 0 and relaxation time 𝜆. The volumetric flow rate of the jet is 𝑄̇ , and the velocity in the jet immediately before the bend is uniform, directed downwards, and equal ( ) to 𝑣 = 𝑄̇ ∕𝜋𝑟20 . Between A and B, the velocity 𝒗 = 𝑣𝑞 , 𝑣𝑟 , 𝑣𝜃 = (𝑣̂ , 0, 0). The time for a fluid particle to pass from A to B is independent of r and 𝜃, and therefore 𝑣̂ (𝑟, 𝜃) = 𝑣(𝑅 + 𝑟 cos 𝜃)∕𝑅 between A and B. This assumption is supported by the particle image velocimetry results of [6], who do not observe shear banding, but note an increase in velocity “closer to the bottom of the jet by geometric effects”. The integral of 𝑣̂ over the cross section at B or A is 𝜋𝑟20 𝑣. The forces the control volume is subject to are those due to gravity, the pressure exerted on the jet by the air layer, and the stresses in the jet at the start and end of the bend. For the system to be in the quasi-steady state, these forces must balance the change in momentum between the start and end of the bend, which may be expressed Δ(𝑚̇ 𝒗) =

∫𝑉

𝜌𝒈𝑑 𝑉 +

∫𝑆

𝑃 (𝑞 , 𝜃)𝑑 𝑺 +

∫𝐴

𝝉 ⋅ 𝑑𝑺 +

∫𝐵

𝝉 ⋅ 𝑑𝑺,

(1)

in which 𝒈 = −9.81𝒆𝒚 is gravity and 𝝉 is the extra stress. All dS are surface elements pointing out of the control volume. For simplicity we rewrite (1) as Δ(𝑚̇ 𝒗) = 𝑭 𝑔𝑟𝑎𝑣 + 𝑭 𝑝 + 𝑭 𝜏𝐴 + 𝑭 𝜏𝐵 ,

(2)

and re-arrange, giving 𝑭 𝑝 = Δ(𝑚̇ 𝒗) − 𝑭 𝑔𝑟𝑎𝑣 − 𝑭 𝜏𝐴 − 𝑭 𝜏𝐵 .

(3)

We now derive expressions for the terms on the right hand side of (3). The mass flux is constant along the jet, and equal to 𝑚̇ = 𝜌𝑄̇ . Noting ) ( that at B, 𝒗𝑩 = −𝑣 𝒆𝒚 cos 𝜙 − 𝒆𝒙 sin 𝜙 , we write ( ) Δ(𝑚̇ 𝒗) = 𝜌𝑄̇ 𝑣 [1 − cos 𝜙]𝒆𝒚 − sin 𝜙𝒆𝒙 . (4) Calculation of the gravity term is straightforward: 𝑭 𝑔𝑟𝑎𝑣 =

∫𝑉

𝜌𝒈𝑑𝑉 =

𝑅𝜙,𝑟0 ,2𝜋

∭0,0,0

𝜌𝒈𝑟𝑑𝜃𝑑𝑟𝑑𝑞 = 𝜌𝒈𝜋𝑟20 𝑅𝜙.

(5)

3.1. Stress at A As y decreases from zero (i.e., as the jet leaves the reservoir), the radius of the jet varies from R0 (the orifice radius) to r0 as the jet undergoes uniaxial extensional strain. We assume that the fluid obeys an Upper Convected Maxwell (UCM) model, given by: ∇

̇ 𝝉 + 𝜆𝝉 = 𝜂0 𝜸,

(6)

where ∇

𝝉=

𝜕𝝉 + (𝒖 ⋅ ∇)𝝉 − (∇𝒖)𝑇 ⋅ 𝝉 − 𝝉 ⋅ ∇𝒖, 𝜕𝑡

(7)

and 𝜸̇ is the strain rate tensor. A number of models of axisymmetric jets of UCM fluids have been derived, based on the assumption that the variation of the solution along the jet is slow relative to the variation across the jet radius. This slowly varying ansatz is used to obtain a system of three coupled ODEs which describe the leading order mode [16,17]. We use a three variable model derived and extensively analysed by Bechtel et al. [16], which may be written as [( ] ) 𝑟̂5 𝑊 𝑒0 𝑚𝑇 − 𝑚𝐻 𝐷𝑒0 𝑑 𝑟̂ = − , (8) 𝑑𝑧 𝐷𝑒0 𝐷 𝑅𝑒0 𝐹 𝑟0 [ ( ) ] 𝐷𝑒 𝑑𝑚𝑇 𝑚𝑇 𝑟̂2 3 3𝑚𝐻 2 =− 𝑟̂ + 2𝑊 𝑒0 − 0 − 𝑟̂ + 2𝑊 𝑒0 , 𝑑𝑧 𝐷𝑒0 𝐷 𝐹 𝑟0 𝑅𝑒0

(9)

[ ( ) ] 𝐷𝑒 𝑑𝑚𝐻 𝑚𝐻 𝑟̂2 3 3𝑚𝑇 2 =− 𝑟̂ + 2𝑊 𝑒0 2 0 − 𝑟̂ + 2𝑊 𝑒0 , 𝑑𝑧 𝐷𝑒0 𝐷 𝐹 𝑟0 𝑅𝑒0

(10)

in which 𝐷 = 𝑟̂3 +

] 2𝑊 𝑒0 [ 𝑇 −𝑚 − 2𝑚𝐻 𝑟̂2 + 2𝑊 𝑒0 . 𝑅𝑒0

(11)

J.R.C. King and S.J. Lind

Journal of Non-Newtonian Fluid Mechanics 273 (2019) 104165

The jet radius 𝑟̂ is non-dimensionalised with respect to the reservoir orifice radius R0 , mT is the integral of the tensile stress over the jet cross section, and mH is the hoop stress resultant. The length scale z is the slowly varying scale 𝑧 = −𝜉𝑦, where 𝜉 = 2𝑅0 ∕𝐻. We0 , Re0 , De0 , and Fr0 are the Weber, Reynolds, Deborah, and Froude numbers, and the subscript 0 indicates they are calculated based on the flow geometry at the reservoir orifice - e.g. 𝐷𝑒0 = 𝜆𝑄̇ ∕𝜋𝑅30 . Note that these non-dimensional numbers are only used in the calculation of F𝜏A . When analysing the results of the force balance, we use non-dimensional numbers based on the geometry of the bend. The system (8)–(10) requires Dirichlet boundary conditions at the reservoir orifice. Whilst we know 𝑟̂ at the reservoir orifice, we do not know the stresses. One option is to assume that the stresses at the reservoir orifice are negligible. However, under this assumption, the model yields a jet with zero stress throughout, and the extensional strain rate of the jet becomes independent of the relaxation time. This is because the term on the right hand side of (6) is dropped in the leading order approximation. Following Feng [18], who noted that upstream boundary conditions in one dimensional models can be problematic, we assume Newtonian stresses at the reservoir orifice. The dimensionless extensional strain rate at the orifice is small (𝜆𝜖̇ < 0.1), and so this assumption appears reasonable. Hence we set 𝑚 𝐻 (𝑧 = 0 ) =

∫

𝜂

𝑊 𝑒0 𝜉 𝑑 𝑣̂ 𝑟𝑑 𝜃𝑑 𝑟 ≈ −2𝑄̇ ( ) 𝜂, 𝑑𝑦 𝐹 𝑟0 1 + 2𝑊 𝑒0 𝐻

(12)

) ( −𝑅𝜙 𝑟 𝑣 sin 𝜃 1 − 𝑒 𝑣𝜆(1+ 𝑅 cos 𝜃) , 𝑅

(15)

and

𝜏𝑞𝑞 = 2𝜂0 𝜆

⎛ ⎡ ⎞ ⎤ −𝑅𝜙 −𝑅𝜙 𝑟 ⎢ ⎥ 𝑣𝜆(1+ 𝑟 cos 𝜃) ⎟ 𝑅𝜙 𝑣2 ⎜ 𝑅 1 − 1 + 𝑒 + 𝜏𝑦𝑦,𝐴 𝑒 𝑣𝜆(1+ 𝑅 cos 𝜃) ( ) ⎜ ⎢ ⎥ ⎟ 2 𝑟 𝑅 ⎜ ⎢ ⎟ 𝑣𝜆 1 + 𝑅 cos 𝜃 ⎥ ⎣ ⎦ ⎝ ⎠ (16)

Note that 𝜏𝑟𝑟 = 0. The final term in (16) is the contribution to the stress due to the extension of the flow near the reservoir orifice, where 𝜏𝑦𝑦,𝐴 = 𝑭 𝜏𝐴 ⋅ 𝒆𝒚 ∕𝜋𝑟20 . We note that 𝝉 ⋅ 𝒆𝒒 = 𝜏𝑞𝑞 𝒆𝒒 + 𝜏𝑞𝑟 𝒆𝒓 + 𝜏𝑞𝜃 𝒆𝜽 , 𝒆𝒒 = −𝒆𝒚 cos 𝜙 − 𝒆𝒙 sin 𝜙 and 𝒆𝒓 = 𝒆𝒙 cos 𝜙 − 𝒆𝒚 sin 𝜙, and that 𝜏 q𝜃 is an odd function of 𝜃, allowing us to write 𝑭 𝜏𝐵 =

𝑟0

∫0

𝜋

∫−𝜋

[ [ ] ] 𝜏𝑞𝑟 cos 𝜙 − 𝜏𝑞𝑞 sin 𝜙 𝒆𝒙 − 𝜏𝑞𝑟 sin 𝜙 + 𝜏𝑞𝑞 cos 𝜙 𝒆𝒚 𝑟𝑑 𝜃𝑑 𝑟. (17)

The exact integration in (17) is non-trivial, but the computational cost of numerical integration is small. We integrate (17) numerically using the mid-point rule, and find a grid-converged solution with a resolution of 𝑁𝑟 = 𝑁𝜃 = 𝑟0 ∕𝛿𝑟 = 2𝜋∕𝛿𝜃 = 50. 4. Discussion

and 𝑇

𝜏𝑞𝜃 = −𝜂0

𝐻

𝑚 = −2𝑚 .

(13)

The integral in (12) is over the cross sectional area of the reservoir orifice, and 𝑣̂ is the streamwise component of velocity in the jet. The term ( ) containing 𝑊 𝑒0 ∕𝐹 𝑟0 1 + 2𝑊 𝑒0 originates from the solution of (8) with zero initial stress. The factor of −2 in (13) arises from the assumption that the variation of 𝑣̂ across the jet radius is small combined with the continuity equation in cylindrical coordinates (leading to 𝑢̂ = 12 𝑟𝑑 𝑣̂ ∕𝑑𝑦 where 𝑢̂ is the radial component of velocity, and hence the diagonal elements of the strain rate tensor 𝜸̇ are 𝛾̇ −𝑦−𝑦 = −2𝑑 𝑣̂ ∕𝑑𝑦 and 𝛾̇ 𝑟𝑟 = 𝑑 𝑣̂ ∕𝑑𝑦). We have investigated the use of an iterative procedure whereby we estimate 𝑑 𝑣̂ ∕𝑑𝑦 at the jet orifice, then obtain an approximate initial stress, solve (8)–(10), then obtain a better estimate of 𝑑 𝑣̂ ∕𝑑𝑦. This approach is substantially more computationally expensive, but has a negligible influence on the final estimate of the stress at A, and we find (12) to be adequate. We solve (8)–(10) numerically with an explicit second order finite difference scheme on a uniform grid of 1000 nodes, yielding a grid-converged solution. For all parameter sets explored, we find that mH ≈ 0 at A, and we simply write 𝑭 𝜏𝐴 = 𝑚𝑇 𝒆𝒚 . An alternative estimate of F𝜏A may be obtained by assuming a simple jet stretch profile, and integrating a linear Maxwell model over the region between the reservoir orifice and A. Details of this approach are given in Appendix B. We find that the use of the UCM or linear Maxwell model give qualitatively similar results for F𝜏A . Through the remainder of the paper we use the UCM model, which is consistent with the model for the flow around the bend, described below. 3.2. Stress at B Next we consider the flow round the bend, between A and B. As the jet has constant radius of curvature round the bend, the strain rate is uniform across the jet (in the 𝑥 − 𝑦 plane), and given by 𝛾̇ 𝑞𝑟 = 𝑅𝑣 cos 𝜃, 𝛾̇ 𝑞𝜃 = −𝑅𝑣 sin 𝜃, with the other elements of 𝜸̇ zero (if the jet radius were not constant, there would be additional terms in 𝛾̇ 𝑞𝑟 and 𝛾̇ 𝑞𝜃 , and the other elements of 𝜸̇ would be non-zero.). Note that the time taken for a fluid particle to travel the length of the bend is given by R𝜙/v. For constant shear rate as in the present case, the UCM model (6) can be solved giving: ) ( −𝑅𝜙 𝑟 𝑣 𝜏𝑞𝑟 = 𝜂0 cos 𝜃 1 − 𝑒 𝑣𝜆(1+ 𝑅 cos 𝜃) , (14) 𝑅

We now have in (4), (5), (8)–(10), and (14)–(17), the means to calculate the terms on the right hand side of (3). We use values and parameters from our experiment to calculate the terms in (3) as follows. Gravity is 𝑔 = 9.81 m/s, the density is 𝜌 = 1023 kg/m3 , the shear viscosity is 𝜂0 = 8.8𝑃 𝑎 ⋅ 𝑠, and the extensional viscosity, calculated from CaBER data, is 𝜂𝑒𝑥𝑡 = 21.1𝑃 𝑎 ⋅ 𝑠. The jet height 𝐻 = 0.117 m, reservoir orifice radius is 𝑅0 = 0.003 m, and volumetric flow rate is 𝑄̇ = 1 × 10−6 m3 ∕s. Values for the geometry of the bend were obtained from photographs of the experiment. The jet radius round the bend is 𝑟0 = 𝑅0 ∕3, and the bend radius is 𝑅 = 5𝑟0 . The bend angle is 𝜙 = 2𝜋∕3. For the fluid in our experiment, experimental values of 𝜆 were obtained by small amplitude oscillatory shear as 𝜆 = 0.056𝑠, and with a capillary breakup extensional rheometer (CaBER) as 𝜆 ∈ [0.1𝑠, 0.14 s] (this discrepancy is expected). We non-dimensionalise all forces with the momentum flux in the jet

𝜌𝑄̇ 2 . 𝜋𝑟20

The Reynolds number based on the length of the bend is 𝑅𝑒𝑏 = 0.39. 4.1. The role of viscoelasticity

Fp is the force that is required to be transmitted through the air layer to bend the jet and balance the system (3). Fig. 4 shows the variation of the forces on the jet as the relaxation time is increased. Note the abscissa shows a Deborah number based on the bend geometry 𝐷𝑒 = 𝜆𝑣∕𝑅𝜙 (discussed in detail later). We see that viscoelasticity plays a key role in the Kaye effect, by altering the force required to bend the jet. This is illustrated in Fig. 5. With increasing values of 𝜆, both the magnitude ( ) and direction of Fp change. We define Θ𝑝 = sin−1 𝑭𝒑 ⋅ 𝒆𝒙 ∕|𝑭𝒑 | as the angle of elevation of Fp above the 𝑥−axis. For 𝜆 = 0 (Newtonian fluids), |𝑭𝒑 | = 2.6, and Fp is directed upwards, and Θ𝑝 = 109◦ , as shown in Fig. 5, panel (a). As 𝜆 increases, |Fp | decreases to a local minimum |𝑭𝒑 | = 1.7 at 𝜆 = 3 ms (𝐷𝑒 = 0.092). Fp then increases to a maximum of |𝑭𝒑 | = 5.0 at 𝜆 = 0.02s (𝐷𝑒 = 0.6), whilst Fp rotates clockwise to a minimum angle of Θ𝑝 = −10.6◦ (see Fig. 5, panel (b)). Over the range of values of 𝜆 obtained for the working fluid in our experiment, |Fp | decreases and slowly rotates anticlockwise, to Θ𝑝 = 50◦ (panels (c) to (e) in Fig. 5). For lower values of |Fp |, the quasi-steady bent jet may be sustained with a lower pressure in the air layer. A greater range of supporting air layers are then permissible, including thicker air layers, which are less likely to rupture and cause the cessation of the effect. For Fp directed

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Journal of Non-Newtonian Fluid Mechanics 273 (2019) 104165

Fig. 4. The forces acting on the control volume as a function of relaxation time. The dashed black line is the change in momentum, the dashed blue line is the force due to gravity, the solid blue line is the stress at the start of the bend, and the solid red line is the stress at the end of the bend. The solid black line is the force that must be exerted on the jet through the air layer to sustain the steady state Kaye effect. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. A schematic diagram showing the direction and magnitude of the force exerted by the heap on the bend for several values of relaxation time 𝜆.

closer to horizontal (as in Fig. 5 panels (b)–(d)), the required asymmetry in the pressure distribution around the bend is such that the pressure near the start of the bend must be lower than near the end. This may also allow for greater entrainment of air, and again enable a thicker air layer to be sustained. The majority of the variation of Fp with 𝜆 is due to change in the stress at B, and therefore 𝑭𝝉𝑩 . For 𝜆 > 0.1s (De > 3), changes in 𝑭𝝉𝑨 are non-negligible, although for 𝜆 in the range of experimental values in the present case (𝜆 ∈ [0.056 s, 0.14s]), these changes are small compared with the variation in 𝑭𝝉𝑩 . The stress in the fluid at B is due to the extension of the fluid as it exits the reservoir, and the strain rate it is subjected to in passing round the bend. In the present case, the latter dominates. For 𝜆 = 0, 𝜏𝑞𝑞 = 𝜏𝑞𝑟 = 0 at B. As 𝜆 is increased, 𝜏 qq becomes non-zero, and 𝒕𝑩 = 𝝉 𝐵 ⋅ 𝒆𝒒 increases, without significant change in orientation for 𝜆 ∈ (0𝑠, 10 s). The effect of this change in tB is that the segment of the jet downstream of the bend exerts a force on the bend, reducing the pressure required in the air layer to balance the system (3). A Deborah number for the flow in the jet round the bend may be defined as 𝐷𝑒 = 𝑄̇ 𝜆∕𝜋𝑅𝜙𝑟20 = 𝑣𝜆∕𝑅𝜙. Note that the exponents in (14)–(16) may be written −1∕[𝐷𝑒(1 + 𝑅𝑟 cos 𝜃)]. As the final term in (16) may be neglected, the location (with respect to 𝜆) of the maximum value of |tB | depends only on De, and is found (by numerical solution) to occur at De ≈ 0.6. Fig. 6 shows the magnitude and direction of Fp plotted against De for a range of bend angles 𝜙, and shows the positions of the maximum of |Fp |, and the minimum of Θp to be independant of 𝜙. In our experiment, 𝐷𝑒 = 1.7 based on the shear relaxation time, and 𝐷𝑒 = 3.07 based on the CaBER relaxation time of 𝜆 = 0.1𝑠. We postulate that the Kaye effect is likely to occur for fluids where De ∈ (0.1, 10), as it is in this region that the direction and magnitude of 𝑭𝝉𝑩 will be most conducive to enabling a thick air layer to be sustained. This indicator alone is of limited use, as De is defined by the geometry of the Kaye effect, which is not known a priori. As noted above, we find that Θp < 0 for some realistic parameter sets (e.g. the present geometry with 𝜆 = 0.02, or the present fluid with 𝜙 ≥ 3𝜋/4), suggesting that the system (3) cannot be in equilibrium, but that the bend in the jet will actually move upwards. We were able to reproduce this effect experimentally, by pouring the working fluid by

hand, such that the flow rate 𝑄̇ was not constant. Under these conditions, we observed the bend in the jet sporadically leap upwards out of the heap, a distance of several jet radii. Observations of the Kaye effect, especially the videos in [6] show that the bend in the jet appears to burrow deeper, until the effect suddently ceases. This behaviour may be viewed as the system described by (3) being unbalanced (i.e., LHS ≠ RHS). As the Kaye effect starts, the jet will burrow deeper, increasing the pressure in the air layer, until (3) is balanced. In some cases, this does not happen before the air layer ruptures, and the Kaye effect ceases. Sometimes the system overshoots, with 𝜙 increasing until the downstream jet interferes with the upstream jet. In our experiment we observed a transient oscillation where the jet was seen to waver (i.e., 𝜙 oscillates around the steady state value). This can be seen in the video included in the supplemental material. Fig. 6 shows a sensitivity of the model to changes in geometry. As 𝜙 increases, Θp at 𝐷𝑒 = 0.6 decreases, whilst the local maximum in |Fp | increases. The sudden change in Θp at De ≈ 0.1 and De ≈ 10 corresponds |Fp | approaching zero. We note that the variation in |Fp | with 𝜙 is small for 𝐷𝑒 = 0.6 over 𝜙 ∈ [2𝜋/3, 4𝜋/5]. Further inspection (numerically) reveals that 𝜕 |Fp |/𝜕 𝜙 ≈ 0 at 𝜙 ≈ 0.73𝜋. This finding supports the observation of the system hunting for a stable state, as changes in geometry about the stable state require only small changes in the force balance. Regarding the choice of upstream boundary condition for the system (8)–(10), larger initial stresses result in larger variations of Fp for large De. However, the trend remains the same, with increasing De giving a reduction in |Fp |. For the present fluid and experimental geometry, the stress at the reservoir orifice plays little role in the force balance around the bend. 4.2. The role of shear thinning behaviour Shear thinning effects may be included in the model, by replacing 𝜂 0 in (14)–(16) with 𝜂𝑒𝑓 𝑓 = 𝜂(𝛾) ̇ , for the value of 𝛾̇ round the bend, according to the experimentally determined Carreau viscosity model given in Appendix A. Defining 𝛼𝜂 = 𝜂𝑒𝑓 𝑓 ∕𝜂0 , we find 𝛼𝜂 = 0.2 and 𝜂𝑒𝑓 𝑓 = 1.8𝑃 𝑎 ⋅ 𝑠. The force balance shown in Fig. 4 is re-calculated, and the results with 𝛼𝜂 = 0.2 are shown in Fig. 7. As 𝜆 is increased, we find that viscoelasticity

J.R.C. King and S.J. Lind

Journal of Non-Newtonian Fluid Mechanics 273 (2019) 104165

Fig. 6. Variation of magnitude and direction of Fp with bend Deborah number for a range of bend angles 𝜙. The red line corresponds to the value of 𝜙 most commonly observed in the experiment. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 7. The forces acting on the control volume as a function of relaxation time, for a shear thinning parameter 𝛼𝜂 = 1∕5.

still plays a role, eventually reducing the magnitude of Fp , and changing its direction. However, the effects of viscoelasticity on the force balance are substantially reduced by the shear thinning behaviour, as the variation in stress at B is reduced by a factor of 𝛼 𝜂 . It has been observed previously [7] that a shear thinning viscosity profile obtained from shear rheometry gives unphysical results when applied to simulations of buckling viscous jets; a finding supported by our own (unpublished) numerical modelling. The flow regimes are substantially different: The rheometer flow involves steady shear, whilst a fluid element in the bent jet only experiences a non-zero deformation rate for a short period of time, as it passes round the bend. Furthermore, although the shear component of the deformation rate tensor is non-zero round the bend in our model, the shear remains zero, such that two particles which are adjacent at A will be adjacent at B. Bonito et al. [7] found that in order to reproduce observed buckling behaviour in jet simulations, they needed a viscosity model which exhibited markedly reduced shear thinning behaviour, with parameters not consistent with those determined from rheometer measurements. We therefore conclude that the viscosity model obtained from shear rheometry is not applicable here. Accordingly, we explore realistic estimates of 𝜂 eff , (and hence 𝛼 𝜂 ), assuming that the shear thinning parameter 𝛼 𝜂 ∈ (0.2, 1]. The magnitude and direction of Fp for a range of 𝛼 𝜂 ∈ [0.2, 1] are shown in Fig. 8. 4.3. The supporting air layer Consideration of the behaviour of the air layer provides some further insight. Starting from the assumption that 𝛿 ≪ r0 , we can approximate the flow within the air layer as a combination of planar Poiseuille and Couette flows, adjusted to account for the non-neglible Knudsen number, following [15]. Neglecting curvature effects, we consider the air

) ( ( [ ]) layer as planar, in the region 𝑞 , 𝑟0 𝜃 ∈ [0, 𝑅𝜙], −𝑟0 𝜓, 𝑟0 𝜓 . Note that we can only approximate 𝜓, the extent to which the air layer wraps around the jet, and in reality 𝜓 is also likely to vary with q, but we assume it to be constant here. The air layer thickness varies with q and 𝜃, and the upper surface of the air layer (the surface of the jet) has veloc( ) ity 𝒗 = 𝑣̂ 𝑟0 , 𝜃 𝒆𝒒 . As 𝜓 is unknown, we limit the analysis to the centre( ) line of the air layer, 𝜃 = 0, on which 𝒗 = 𝑣̂ 𝑟0 , 0 𝒆𝒒 = 𝑉 𝒆𝒒 . Following the derivation in [15] we obtain expressions for the mean velocity in the air layer, given by: ⎧ ⎫ ( ) ⎪ 𝑉 𝛿 2 𝜕𝑝 6𝐾𝑛 ⎪ 𝒗̄ = ⎨ ( 𝒆𝒒 (1 + 𝛼𝐾𝑛) 1 + ( ) )− 12𝜇0 𝜕𝑞 1 + 𝐾𝑛 ⎬ ⎪ 2 1 + 2 2−𝜎𝑣 𝐶𝑚 𝐾𝑛 ⎪ 𝜎𝑣 ⎩ ⎭ ( ) 𝜕𝑝 𝛿2 6𝐾𝑛 − (18) 𝒆 , ( ) (1 + 𝛼𝐾𝑛) 1 + 12𝜇0 𝜕 𝑟0 𝜃 1 + 𝐾𝑛 𝜽 where 𝜇 0 is the dynamic viscosity of air at atmospheric pressure, 𝛼 is a rarefaction correction parameter, taken as 1.3 in this work, and 𝜎 v is the tangential accommodation coefficient, which describes the statistical mean tangential momentum transfer for molecular interactions between ( ) the gas and the liquid surface. 𝜎 v is defined as 𝜎𝑣 = 𝜏𝑖 − 𝜏𝑟 ∕𝜏𝑖 , where 𝜏 i and 𝜏 r are tangential momenta of incident and reflected molecules respectively. Values of 𝜎𝑣 = 1 and 𝜎𝑣 = 0 correspond to purely diffuse and specular molecular interactions, respectively. In the following, we investigate two values of accommodation coefficient - 𝜎𝑣 = 1 and 𝜎𝑣 = 0.3. We set 𝐶𝑚 = 1.11, which corresponds to a first-order slip boundary condition. Note that (18) differs from the result of [15], in that we have included the effects of rarification in the mean Couette velocity (the first term on the RHS), to account for the stationary frame of reference. We obtain a modified slip-corrected Reynolds equation by constraining

J.R.C. King and S.J. Lind

Journal of Non-Newtonian Fluid Mechanics 273 (2019) 104165

Fig. 8. Variation of magnitude and direction of Fp with relaxation time for a range of shear thinning parameter 𝛼 𝜂 , with 𝜙 = 2𝜋∕3.

the divergence of the mass flux in the air layer to be zero: ( ) ∇ ⋅ 𝜌𝑎 𝒗̂ 𝛿 = 0,

(19)

where 𝜌a is the air density (proportional to pressure, in the case of uni𝜕𝑝 𝜕𝛿 form temperature). Noting that on the line 𝜃 = 0, 𝜕𝜃 = 𝜕𝜃 = 0, we write the modified slip-corrected Reynolds equation as ⎧ ⎫ ( ) 𝑝 Λ𝑅 𝛿 𝜕 ⎪ 6𝐾𝑛 ⎪ 3 𝜕𝑝 (1 + 𝛼𝐾𝑛) 1 + ( ( ) ) − 𝑝𝛿 𝜕𝑞 ⎨ 𝜕𝑞 1 + 𝐾𝑛 ⎬ ⎪ 1 + 2 2−𝜎𝑣 𝐶𝑚 𝐾𝑛 ⎪ 𝜎𝑣 ⎩ ⎭ ( ) 2 𝜕 𝑝 6𝐾𝑛 +𝑝𝛿 3 ( ) (1 + 𝛼𝐾𝑛) 1 + = 0, 2 1 + 𝐾𝑛 𝜕 𝑟0 𝜃

(20)

where Λ𝑅 = 6𝜇0 𝑉 . Eq. (20) is equivalent to equation 20 in [15], with the addition of a source term describing the transverse mass flux. Eq. (20) is nonlinear in two unknowns - p and 𝛿. In order to solve (20) we require additional assumptions. Firstly, we estimate 𝜓 = 𝜋∕2, which is a reasonable estimate based on the images in Fig. 4 of [6], where the extent of the air layer can be seen by the strings of bubbles observed as the air layer collapses. Along the line 𝜃 = 0, the condition 𝜕 𝑝∕𝜕 𝜃 = 0 is required by symmetry considerations, whilst on 𝜃 = 𝜓 , 𝑝(𝑞, 𝜓 ) = 𝑝𝑎𝑚𝑏 and 𝜕 𝑝∕𝜕 𝜃 = 0, where pamb is the ambient pressure in the vacuum chamber. These conditions allow us to assume a form for the variation of pressure with 𝜃: ( ( )) 𝑝 − 𝑝 𝑐 (𝑞 ) 𝜃𝜋 𝑝(𝑞 , 𝜃) = 𝑝𝑎𝑚𝑏 + 𝑎𝑚𝑏 1 + 𝑐𝑜𝑠 , (21) 2 𝜓 where pc is the pressure in the air layer underneath the centreline of the jet (i.e. along the line 𝜃 = 0). Eq. (21) can be differentiated twice to obtain the necessary second derivative in the final term in (20), and will be used in the integration of the resulting pressure field to determine Fp due to the air layer. Finally, as we cannot solve (20) for both 𝛿 and p, we prescribe one, and solve for the other. We solve (20) using a finite difference scheme, with 500 nodes along the length of the air layer. Prescribing the pressure in the air layer to match the hydrostatic pressure variation in the heap, and solving for 𝛿, we calculate the relative reduction in thickness between the start and end of the air layer: ( ) Δ𝛿% = 100 𝛿0 − 𝛿𝑅𝜙 ∕𝛿0 , where the subscripts 0 and R𝜙 indicate the start and end of the bend respectively. Varying the initial air layer thickness 𝛿 0 , we find that for thicker air layers Δ𝛿 % is larger. Varying the ambient pressure pamb we find that Δ𝛿 % increases with decreasing ambient pressure. However, the variation of 𝛿 with q is small, as the mass flux due to the pressure gradients (Poiseuille flow contribution) is much smaller than the mass flux due to the motion of the jet (Couette flow contribution). For 𝛿0 = 0.5𝜇𝑚 (the thickness estimated by Lee et al. [6]), the reduction in thickness along the air layer is Δ𝛿% = 0.154% for 𝑝𝑎𝑚𝑏 = 105 𝑃 𝑎, increasing to 0.4% for 𝑝𝑎𝑚𝑏 = 3 × 104 𝑃 𝑎. For a smaller value of 𝛿 0 , the relative change is smaller. For 𝛿0 = 10μm, nearing the limit of validity of the model (i.e. where 𝛿 ≪ r0 ceases to be true),

Fig. 9. Pressure variation along the length of the air layer predicted by lubrication theory (Eq. (20)) for a range of thicknesses 𝛿 0 , for 𝑝𝑎𝑚𝑏 = 105 𝑃 𝑎.

Δ𝛿% = 17.7% for 𝑝𝑎𝑚𝑏 = 105 𝑃 𝑎, rising to 19.1% for 𝑝𝑎𝑚𝑏 = 3 × 104 𝑃 𝑎. Although sensitivity to ambient pressure change appears low, clearly a smaller ambient pressure causes a greater relative decrease in air layer thickness along its length. Next, we prescribe a linear variation of 𝛿 with q, such that 𝛿 reduces in thickness by 1% over the length of the air layer. This relative variation in 𝛿 yields pressure distributions of the same order of magnitude as the hydrostatic pressure variation in the heap: a necessary condition for the quasi-steady assumption to hold. Fig. 9 shows the pressure variation within the air layer as predicted by lubrication theory. For large values of 𝛿 the pressure distribution takes a sinusoidal form (with decreasing amplitude as 𝛿 is increased), whilst for increasingly thin air layers, the pressure distribution tends towards a half-wavelength sawtooth profile. The location at which the air layer ruptures is likely to be close to the peak pressure in the air layer, where the interfacial stresses will be greatest. The peak in pressure towards the downstream end of the bend seen in Fig. 9 agrees with the observations in [6], where air layer rupture was seen to occur near the downstream end of the bend. Fig. 10 shows the variation of |Fp | (calculated from lubrication theory) with pamb for a range of 𝛿 0 . As 𝛿 0 is reduced there is a transition from a logarithmic dependence of |FP | on pamb , to a linear dependence. This trend holds for different variations of 𝛿 and choices of 𝜓. We find the same relations (logarithmic and linear) for a range of accommodation coefficients 𝜎 v . Lower values of 𝜎 v give larger gradients 𝜕 |Fp |/𝜕 pamb . Values of 𝜎 v for air-shampoo interfaces are unavailable, although the closest analogue available in the literature [19] - the interface between various gases and oil-films - suggests values typically of 𝜎 v ∈ (0.8, 1).

J.R.C. King and S.J. Lind

Journal of Non-Newtonian Fluid Mechanics 273 (2019) 104165

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments We are grateful to the Leverhulme Trust for funding this work via Research Project Grant RPG-2017-144. We would also like to thank Professor Rob Poole of the University of Liverpool for several useful discussions, and, along with Dr Henry Ng, for characterising the rheology of our shampoo. We thank several anonymous reviewers for some very insightful comments and suggestions. Fig. 10. Variation of magnitude of Fp with ambient pressure pamb predicted by lubrication theory (Eq. (20)) for a range of thicknesses 𝛿, for 𝜎𝑣 = 1 (solid lines), and 𝜎𝑣 = 0.3 (dashed lines).

Importantly, for a range of thicknesses and ambient pressures, lubrication theory predicts quantitatively similar values of |Fp | as (3). From Fig. 6, we expect the Kaye effect to occur for moderate De with |Fp | values near and above 1. These are values predicted by lubrication theory for ambient pressures at and above the pcrit ≈ 32kPa cut-off, for realistic air layer thicknesses (e.g. 𝛿0 = 0.5μm), as seen in Fig. 10. It is clear that lubrication theory alone cannot fully explain the value of pcrit found in our experiment, suggesting that the tipping point apparent in Fig. 2 arises not only because of the reduced load capacity of the air layer, but also due to changes in air entrainment with reducing ambient pressure. This is supported by the video included in the supplementary material, which shows that as pressure is reduced, the Kaye effect becomes more sporadic, with occurances of the effect becoming shorter lived. Below pcrit we see no Kaye effect at all - the jet simply coils, and the coiled heap subsides. Greater insight into entrainment and the formation of the air layer could be provided by further numerical work. However, such work is beyond the scope of this study; the development of a high-fidelity, multi-resolution, multi-phase viscoelastic numerical scheme remains a non-trivial task, and is an ongoing area of research for the authors.

5. Conclusions We have conducted an experiment and found that the Kaye effect does not occur in a vacuum. This observation supports the theory that during the Kaye effect the jet slides on a thin air layer, and demonstrates that the theory that the jet slides on a shear-thinning slip layer is incomplete. Given the existence of the air layer, we show via control volume analysis that viscoelasticity is key. Viscoelasticity alters the stress-response of the jet to the strain it is subject to in passing round the bend, changing the magnitude and direction of the force required to bend the jet. We find that with increasing degree of shear thinning behaviour, this viscoelastic response is reduced. Shear thinning behaviour in the absence of elasticity does not alter the force balance relative to a Newtonian fluid. By reducing the air pressure force required and, in effect co-supporting the bend, viscoelasticity enables more air to be entrained, and a thicker air layer to be created for fluids with moderate relaxation times, thereby increasing the likelihood and duration of jet slippage (i.e. the Kaye effect). We find this most apparent for Deborah number based on the bend travel-time being De ∈ (0.1, 10). Air layer force predictions due to control volume analysis and micro-scale lubrication theory are quantitatively similar. In certain circumstances, the jet can be entirely supported by the elastic stresses caused by bending, and the bend lifts away from the heap, a phenomenon we have observed in our experiments.

Appendix A. Details of the experiment The vacuum chamber was cylindrical with diameter 300 mm and height 350 mm, with a transparent lid, connected to a 186W vacuum pump. Within the vacuum chamber we fixed a reservoir 150 mm diameter, filled to a depth of 55 mm. The reservoir had an orifice of 6 mm diameter at the centre of the base. Below the reservoir was an acrylic box, with transparent sides, and a base angled at approximately 15 degrees from horizontal. The height from the reservoir orifice to the plane of impact was initially 117 mm. The primary working fluid was a widely available commercial shampoo, chosen because it was the first shampoo we tested which exhibited the Kaye effect for jet heights which fitted within the vacuum chamber. The zero shear viscosity of the working fluid was measured as 𝜂0 = 8.8𝑃 𝑎 ⋅ 𝑠, and the density 𝜌 = 1023 kgm−3 . The widely used Carreau viscosity model ( )( ) 𝑛−1 𝜂𝑒𝑓 𝑓 = 𝜂∞ + 𝜂0 − 𝜂∞ 1 + 𝑎2 𝛾̇ 2 2 ,

(A.1)

was fitted to steady shear rheometry data, yielding parameters 𝜂∞ = 10−3 𝑃 𝑎 ⋅ 𝑠, 𝑎 = 0.1𝑠 and 𝑛 = 0.12. The viscosity profile is shown in Fig. A.11. A relaxation time of 𝜆 = 0.056 s was calculated by conducting a Small Amplitude Oscillatory Shear (SAOS) test and taking the reciprocal of the angular frequency for which the elastic and viscous moduli were equal. The SAOS data are shown in Fig. A.12. A second estimate of the relaxation time was calculated using a capillary breakup extensional rheometer (CaBER), giving 𝜆 ∈ [0.1s, 0.14s]. We note that the parameter a in the Carreau model, which is a relaxation time, is closer in value to the relaxation time obtained from extensional rheometry than that obtained from shear rheometry. Surface tension was calculated using the

Fig. A.11. Steady shear rheometry data for the working fluid: variation of viscosity with shear rate. Data were obtained over three sweeps of shear rate (indicated by different symbols). The solid line is the Carreau model fitted to the data.

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Journal of Non-Newtonian Fluid Mechanics 273 (2019) 104165

distance 𝛼R0 , as ( ) 𝜖𝜋𝑅 ̇ 30 𝛼 + 𝑄̇ 𝑡0𝛼 = 𝜖̇ ln , 𝑄̇

(B.2)

and the time for the fluid to pass from −𝑦 = 𝛼𝑅0 to A is 𝑡𝛼𝐴 = ( ) 𝐻 − 𝛼𝑅0 ∕𝑣. We assume that the fluid in extension obeys a simple 1D Maxwell model, given by 𝜏𝑦𝑦 + 𝜆𝜕 𝜏𝑦𝑦 ∕𝜕 𝑡 = 𝜂𝑒𝑥𝑡 𝜖,̇ and 𝜏𝑥𝑥 = 𝜏𝑥𝑦 = 0, where 𝜂 ext is the apparent extensional viscosity of the fluid determined from CaBER measurements. For this model, the stress response to an impulse in extensional strain rate (i.e. 𝜖̇ = 𝛿(𝑡), where 𝛿(t) is the Dirac delta function) is given by 𝑡 𝜂 𝜏𝑦𝑦,𝛿 = 𝑒𝑥𝑡 𝑒− 𝜆 . (B.3) 𝜆 Fig. A.12. Small amplitude oscillatory shear rheometry data: variation of bulk and loss moduli with angular frequency 𝜔. The cross-over occurs at approximately 𝜔 = 17.8𝑟𝑎𝑑∕𝑠.

By taking the convolution of (B.3) with the extensional strain rate given by (B.1), we obtain the extra stress at A: ( ) 𝑡 −𝑡0𝛼 𝛼𝐴 𝜏𝑦𝑦,𝐴 = 𝜂𝑒𝑥𝑡 𝜖̇ 1 − 𝑒 𝜆 𝑒 𝜆 (B.4)

capillary rise method at atmospheric pressure as 𝜎 = 0.02 N/m. Observations of sessile droplets as the vacuum chamber was evacuated showed no visible change in droplet shape, and thus surface tension was assumed constant throughout. From the known surface tension, and CaBER filiament thinning rate, the effective extensional viscosity 𝜂𝑒𝑥𝑡 = 21.08𝑃 𝑎 ⋅ 𝑠 was calculated. For the 6mm diameter orifice, the jet had a volumetric flow rate of 1 ml/s. The reservoir orifice diameter was subsequently varied from 3 mm to 10 mm, giving a range of flow rates between 0.2 ml/s and 6 ml/s, whilst the height of the reservoir was varied between 115 mm and 150 mm. The flow rate did not vary significantly over the course of a single evacuation-repressurisation cycle. Images of the Kaye effect were captured using a Sony DSC RX100 digital camera directed through the vacuum chamber lid, to a mirror inclined at approximately 45∘ , at a resolution of 1920 × 1080 pixels and a frame rate of 25fps. Pressure readings were taken using the vacuum chamber’s analogue pressure gauge at intervals of 5 s over several runs, and were found to be repeatable to within measurement error. An analytic expression for the pressure as a function of time (after switching the pump on) was fitted to the pressure readings. This expression gave an estimate of the pressure at each time frame in the videos. The proportion of the time the Kaye effect occured (as in Fig. 2) was calculated by downsampling the videos to 5fps, and counting frames in which the Kaye effect occurred, in 4 second bins. The experiment was conducted in a lab with ambient temperature between 18∘ C and 20∘ C. Temperature measurements in the vacuum chamber showed slight drop in temperature during evacuation, of approximately 2∘ C.

The force due to the extra stress at A is therefore ( ) 𝑡 −𝑡0𝛼 𝛼𝐴 𝑭 𝜏𝐴 = 𝝉 ⋅ 𝑑 𝑺 = 𝜋𝑟20 𝜂𝑒𝑥𝑡 𝜖̇ 1 − 𝑒 𝜆 𝑒 𝜆 𝒆𝒚 . ∬𝐴

Appendix B. Linear maxwell model for the falling jet [ ] We assume that in the region −𝑦 ∈ 0, 𝛼𝑅0 the extensional strain rate is constant, with value 𝑣− 𝜖̇ =

𝑄̇ 𝜋𝑅20

𝛼𝑅0

,

(B.1)

and that the extensional strain rate is zero elsewhere. The local vertical ̇ velocity of the jet can be written − 𝑑𝑦 = 𝑄2 − 𝜖𝑦, ̇ which can be inte𝑑𝑡 𝜋𝑅0

grated to find an expression for the time taken for the fluid to fall a

(B.5)

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